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Getting Tietze Representation in Original Group

asked 2023-06-29 00:29:52 +0200

thethinker gravatar image

I've got a free group with relators:

sage: H.<a,b,c,d>=FreeGroup();H
Free Group on generators {a, b, c, d}
sage: Y=[[1,2],[3,4]]
sage: G=H/[Y[i] for i in range(2)];G
Finitely presented group < a, b, c, d | a*b, c*d >
sage: I=G.simplification_isomorphism();I
Generic morphism:
  From: Finitely presented group < a, b, c, d | a*b, c*d >
  To:   Finitely presented group < a, c |  >
  Defn: a |--> a
        b |--> a^-1
        c |--> c
        d |--> c^-1

Cool. Now I want to get the Tietze representation. First, under the isomorphism:

sage: G([4])
sage: I(G(([4])))
sage: I(G([4])).Tietze()

Sure, since the index of the element is the second in the list under the isomorphism. But, I want to be able to grab the element under the original free group - that is, I want (-3). Something that I think should work:

sage: x=I(G([4]));x
sage: H(x).Tietze()

Right? "What is the Tietze representation of the word 'x' in the group 'H'"....but no, still under the isomorphism. Even worse, it looks like it's like, using the Tietze representaton to do something that looks very wrong:

sage: H(I(G([4])))

That's very confusing, but at least clearly not what I want. I can do this if I know the word ahead of time:

sage: y=c^-1
sage: y.Tietze()

But the point of the code I'm trying to write is to determine these words under the isomorphism. Anyone know how to do something like my second code block, but with the response (-3)?

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answered 2023-07-23 00:52:57 +0200

dan_fulea gravatar image

This answer shows why we have the irritating, counterintuitive results in between, and tries to give a solution for a "correct lift". Let us consider a definition of $H,G$ in a more common manner.

H.<a,b,c,d> = FreeGroup()
G = H.quotient([a*b, c*d])
J = G.simplification_isomorphism()
C = J.codomain()

print(f"H is {H}")
print(f"G is {G}")
print(f"C is {C}")
print(f"J is {J}")
print(f"Is G the domain of definition of J? {J.domain() == G}")
print(f"Is C the codomain of J? {J.codomain() == C}")

The above code delivers the following information:

H is Free Group on generators {a, b, c, d}
G is Finitely presented group < a, b, c, d | a*b, c*d >
C is Finitely presented group < a, c |  >
J is Generic morphism:
  From: Finitely presented group < a, b, c, d | a*b, c*d >
  To:   Finitely presented group < a, c |  >
  Defn: a |--> a
        b |--> a^-1
        c |--> c
        d |--> c^-1
Is G the domain of definition of J? True
Is C the codomain of J? True

Now let us say some words on the way we can represent an element of $H$. Consider the element $abcd\bar a\bar b\bar c\bar d$. (A bar denotes the inverse, it is simpler to type it.)

One possibility is to type:

sage: a * b * c * d * a^-1 * b^-1 * c^-1 * d^-1
sage: _.parent()
Free Group on generators {a, b, c, d}

But an other way is, using some sage in between representation as a tuple:

sage: H( (1, 2, 3, 4, -1, -2, -3, -4) )
sage: (a * b * c * d * a^-1 * b^-1 * c^-1 * d^-1).Tietze()
(1, 2, 3, 4, -1, -2, -3, -4)

And the tuple plugged in in the first line above is accepted, the entries are numbers, these numbers are insensible w.r.t. the group we are using for them. It may be $H$, or $G$, or $C$, the main point is to not use numbers bigger (in modulus) then the number of generators. So we can also write

sage: G( (1, 2, 3, 4, -1, -2, -3, -4) )

and there is no error (but also no simplification). But we cannot write

sage: C( (1, 2, 3, 4, -1, -2, -3, -4) )
AttributeError                            Traceback (most recent call last)

and there is a long error message following.

We consider now $d\in H$. It is mapped to $d$ in the quotient. To distinguish the two $d$'s in code we write d or H(d) for the element $d\in H$, and G(d)for the element $d$ in the quotient $G$. Then $J$ maps $G(d)$ into...

sage: J(G(d))
sage: C( (-2,) )
sage: # don't try J(G(d)) == C( (-2,) )

So $J(d)=\bar c$, an equality inside $C$. Now we would like to see this element correctly lifted inside $H$. The "obvious" $H(C(-2))$ is taking the "Tietze-tuple" (-2, ) and does the best to get an element inside $h$ for it.

To have the right lift, we may want to define a morphism from $C$ to $H$.

sage: j = C.Hom(H)( (a, c) )
sage: j(C(a))
sage: j(C(c))

The definition of the morphism is as follows, in human words. Consider some homomorphism from $C$ to $H$, this is the part with C.Hom(H). Which one? The one that maps the generators of $C$, in their given order to the elements of the tuple (a, c), also in this order. And with these objects...

  • my expectation was that $j(J(G(d)))$ should work. No, it is not working. (The reason is maybe that somehow as above, J(G(d)) == C( (-2,) ) evaluates to False.)
  • so let us oblige sage to do the right thing by inserting the two inverse operations of getting the Tietze representations, and building the $C$-element for such a representation:

    sage: j( C( J(G(a)).Tietze() ) )
    sage: j( C( J(G(b)).Tietze() ) )
    sage: j( C( J(G(c)).Tietze() ) )
    sage: j( C( J(G(d)).Tietze() ) )
    sage: j( C( J(G(a * b * c * d * a^-1 * b^-1 * c^-1 * d^-1)).Tietze() ) )

I'm sorry, i don't have a better solution...

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Asked: 2023-06-29 00:29:52 +0200

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Last updated: Jul 23 '23